All eyes will be on President Trump this week as the Senate begins its immigration debate, with Republicans hoping the White House will play an active role in ruling out proposals he won’t accept, making it likelier he will get a bill he can sign.

The first test vote will occur Monday as senators vote to bring up a placeholder piece of legislation and then begin to offer their proposals as amendments.

How that will play out remains to be seen, but senators were pushing over the weekend to get plans ready.

The latest is an offer from Republican Senate leaders, led by Sens. Charles E. Grassley of Iowa and John Cornyn of Texas, which enshrines Mr. Trump’s four-point framework with an enforcement-heavy proposal. It would couple citizenship rights for perhaps 1.8 million Dreamers with some of the biggest enforcement changes in two decades.

“This is a rare opportunity to fix a real problem and protect the country in a thoughtful and compassionate way,” Mr. Grassley said. “We simply have to correct the loopholes in current law that allow dangerous criminals to enter and remain at large in our country.”

He said their bill has the backing of Mr. Trump — which is more than any of the other proposals being batted about in the Senate.

White House officials were on Capitol Hill last week coordinating strategy, with legislative liaison Marc Short seen shuttling in and out of Senate Republican offices.

But Mr. Trump’s willingness to declare plans dead on arrival could determine the outcome of the debate.

After a year in office, this White House has been reticent about using the V-word, having issued more than 50 official statements of policy on bills and amendments over the past year without once explicitly threatening a veto.

Some informal threats have been made, though, including during the tax cut debate when the president suggested he would reject any plan that subjected 401(k) retirement savings to new taxes.

That threat was effective, putting to rest any thoughts of targeting 401(k) plans.

Immigration enforcement backers hope Mr. Trump will play the same role in the Dreamer debate.

“It’s all up to Trump now,” said Rosemary Jenks, government relations manager at Numbers USA, which advocates for stricter immigration controls.

But that has done little to slow them down. The two men said they will offer a version of their plan — which would couple a generous amnesty with a small down payment on the border wall, tweaks to the timing of family migration and a shuffling of the diversity lottery’s visas — as part of the debate.

Meanwhile Sen. John McCain, Arizona Republican, is teaming up with Sen. Christopher A. Coons, Delaware Democrat, to propose an even more generous amnesty with less security. They would couple legalization for perhaps 3 million illegal immigrants with a study of border security, and promises of action by 2021.

Several sources familiar with the negotiations said Sens. Mark R. Warner, Virginia Democrat, and Johnny Isakson, Georgia Republican, are working on what amounts to a fallback plan, offering to renew the Obama-era Deferred Action for Childhood Arrivals program along with three years of funding for Mr. Trump’s border wall plans.

Neither senator’s office returned messages seeking comment.

A White House official signaled that proposal still fell short, saying partial construction of the border wall wouldn’t be enough to combat a potential surge in border jumpers enticed by the prospect of an amnesty.

Ms. Jenks said each of the plans had the same flaw: They embraced Mr. Trump’s offer of citizenship for Dreamers but rejected most of his border security and policy changes.

“All of the cabals going on in the Senate, except the Grassley-Cornyn one, are totally ignoring everything President Trump laid out, except the 1.8 million,” Ms. Jenks said.

Her group is backing a proposal sponsored by House Judiciary Committee Chairman Bob Goodlatte, Virginia Republican, that would couple a legalization for hundreds of thousands of Dreamers — though not citizenship rights — with major policy that changes they hope would finally solve illegal immigration.

Among the changes would be requiring businesses to check new employees’ work status through the E-Verify system, cracking down on sanctuary cities and promoting more cooperation in deportations.

The White House said it liked that plan — though Mr. Trump’s proposal is far more generous toward Dreamers.

And the White House was closely involved in writing the new Grassley-Cornyn plan, which couples legalization of 1.8 million Dreamers with $25 billion to build a border wall, an end to the catch-and-release policies that free illegal immigrants into the interior of the U.S., faster deportations for visitors who overstay their permits, and permanent authorization of the E-Verify system that some businesses use to check their workers’ legal status.

The Republican proposal would also eliminate the Diversity Visa Lottery and reuse those visas to lower the immigration backlog.

The senators also called for much stricter limits on extended family members who can be sponsored for future immigration — though parents would still be allowed to come on generous visitor visas. The stricter limits would be delayed until after the current family-based immigration backlog is cleared.

The plan both expands on Mr. Trump’s security proposals — E-Verify and a crackdown on repeat illegal immigrants were not explicitly part of the president’s framework — while making changes to ensure that overall legal immigration numbers aren’t reduced. That is something most Democrats and Republicans on Capitol Hill have said is non-negotiable.

Still, the price tag of the wall and the other security changes are likely to be difficult to swallow for Democrats, who have indicated no willingness to crack down on current illegal immigrants as part of the ongoing negotiations.

In the House, meanwhile, Republican leaders have encouraged the Goodlatte plan but have not put much muscle behind it. For now, Speaker Paul D. Ryan and other Republican leaders seem content to watch the Senate debate and see what can clear that chamber before deciding their own path.

Mr. Ryan, however, has repeatedly said he will not bring anything to the House floor that Mr. Trump cannot sign.

“We’re not going to bring immigration legislation through that the president doesn’t support,” Mr. Ryan said.

Senators who have regularly battled Mr. Trump, however, are hoping to overwhelm him with a show of support for legislation that doesn’t live up to his framework on issues such as cutting chain migration.

“I still think that if we put a good bill to the president, that has the support of 65, 70 members of the Senate, that the president will accept it and the House will like it as well,” Sen. Jeff Flake, Arizona Republican, told NBC’s “Meet the Press.”

“By definition, if we can get something with support of 65 to 70 senators or maybe more, it’s going to be a good, broad bill that will address, I believe, the concerns that the president has outlined, but also take into account the things that we need for our economy going forward.

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